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Identifying and tracking an unknown linear system from observations of its inputs and outputs
is a problem at the heart of many different applications. Due to the complexity and
rapid variability of modern systems, there is extensive interest in solving the problem with
as little data and computation as possible.
This thesis introduces the novel approach of reducing problem dimension by exploiting
statistical structure on the input. By modeling the input to the system of interest as a
graph-structured random process, it is shown that a large parameter identification problem
can be reduced into several smaller pieces, making the overall problem considerably simpler.
Algorithms that can leverage this property in order to either improve the performance
or reduce the computational complexity of the estimation problem are developed. The first
of these, termed the graphical expectation-maximization least squares (GEM-LS) algorithm,
can utilize the reduced dimensional problems induced by the structure to improve the accuracy
of the system identification problem in the low sample regime over conventional methods
for linear learning with limited data, including regularized least squares methods.
Next, a relaxation of the GEM-LS algorithm termed the relaxed approximate graph
structured least squares (RAGS-LS) algorithm is obtained that exploits structure to perform
highly efficient estimation. The RAGS-LS algorithm is then recast into a recursive
framework termed the relaxed approximate graph structured recursive least squares (RAGSRLS)
algorithm, which can be used to track time-varying linear systems with low complexity
while achieving tracking performance comparable to much more computationally intensive
methods.
The performance of the algorithms developed in the thesis in applications such as channel
identification, echo cancellation and adaptive equalization demonstrate that the gains admitted
by the graph framework are realizable in practice. The methods have wide applicability,
and in particular show promise as the estimation and adaptation algorithms for a new breed
of fast, accurate underwater acoustic modems.
The contributions of the thesis illustrate the power of graphical model structure in simplifying
difficult learning problems, even when the target system is not directly structured.

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Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 2016

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